Nat. Hazards Earth Syst. Sci., 6, 903–909, 2006 www.nat-hazards-earth-syst-sci.net/6/903/2006/ Natural Hazards © Author(s) 2006. This work is licensed and Earth under a Creative Commons License. System Sciences On the singular values decoupling in the Singular Spectrum Analysis of volcanic tremor at Stromboli R. Carniel1, F. Barazza1, M. Tarraga´ 2, and R. Ortiz2 1Dipartimento di Georisorse e Territorio, Universita` di Udine, 33100 Udine, Italy 2Departamento de Volcanolog´ıa, Museo Nacional de Ciencias Naturales, CSIC, Madrid, Spain Received: 6 December 2005 – Revised: 11 October 2006 – Accepted: 11 October 2006 – Published: 16 October 2006 Abstract. The well known strombolian activity at Strom- this kind of events, and therefore the possible appearance boli volcano is occasionally interrupted by rarer episodes of of recordable precursors. Talking about timescales, (Ripepe paroxysmal activity which can lead to considerable hazard et al., 2002) identified two degassing and explosive regimes for Stromboli inhabitants and tourists. On 5 April 2003 a at Stromboli, linked to the fresh gas-rich magma supply rate, powerful explosion, which can be compared in size with the that alternate on a 5–40 min timescale. At a longer timescale, latest one of 1930, covered with bombs a good part of the (Carniel and Di Cecca, 1999) identified days-to-weeks-long normally tourist-accessible summit area. This explosion was dynamical phases, sometimes separated by abrupt transi- not forecasted, although the island was by then effectively tions. Talking about possible precursors, (Carniel and Iacop, monitored by a dense deployment of instruments. After hav- 1996b) showed that paroxysmal phases are sometimes pre- ing tackled in a previous paper the problem of highlight- ceded by increasing lower frequency content in the tremor. ing the timescale of preparation of this event, we investigate Due to the complexity of the physical processes involved, here the possibility of highlighting precursors in the volcanic a stochastic (Jaquet and Carniel, 2001, 2003) or dynamical tremor continuously recorded by a short period summit seis- approach (Carniel and Di Cecca, 1999; Carniel et al., 2003) mic station. We show that a promising candidate is found is often a more appropriate choice for short-to-medium term by examining the degree of coupling between successive sin- forecasts aimed e.g. to schedule the tourist excursions when gular values that result from the Singular Spectrum Analysis volcanic hazard is lower. of the raw seismic data. We suggest therefore that possible On 28 December 2002, an effusive eruption (Alean anomalies in the time evolution of this parameter could be et al., 2006) marked the most significant effusive episode indicators of volcano instability to be taken into account e.g. since 1985–1986. This induced considerable morpholog- in a bayesian eruptive scenario evaluator. Obviously, further ical changes before its end on 21–22 July 2003 (Calvari, (and possibly forward) testing on other cases is needed to 2003) and showed the most energetic associated explosive confirm the usefulness of this parameter. event at 07:12 GMT of 5 April 2003. The event started with a reddish ash emission related to collapses within the craters, soon replaced by darker juvenile material from NE 1 Introduction Crater and followed by similar emissions from SW Crater which mushroom-shaped dark cloud rose about 1 km above Stromboli is the prototype for strombolian activity, observed the summit (Calvari, 2003). Bombs, ash, and blocks af- since at least 3–7 A.D. (Rosi et al., 2000). Occasionally, fected most the Western sector and the village of Ginostra; paroxysmal phases are observed (Barberi et al., 1993; Jaquet the volcano top above 700 m a.m.s.l. was completely covered and Carniel, 2003) involving an additional kind of magma, by bombs, including meter-sized ones (Alean et al., 2006), low-porphyritic and volatile-rich (Francalanci et al., 2004) and scientific equipment, including the seismic station of the that apparently plays a major role in the development of such University of Udine, was damaged or destroyed. In a previ- phases. This involvement of a different magma suggests that ous paper, Carniel et al. (2006) tackled the problem of high- there may be a sufficiently long timescale of preparation for lighting the timescale of preparation of this event, showing that the volcanic tremor recorded continuously by a short Correspondence to: R. Carniel period seismic station is sufficiently informative to derive ([email protected]) this kind of information. In fact, non-stationarity is a key Published by Copernicus GmbH on behalf of the European Geosciences Union. 904 R. Carniel et al.: Singular values decoupling characteristic of the tremor (Konstantinou and Schlindwein, This is by definition a Hankel matrix, as it is Xi,j =xi+j−1. 2002), as it offers the possibility of monitoring changes in pa- The size of the embedding window m (number of columns rameters derived from an experimental time series and their of the trajectory matrix) should be sufficiently large to cap- possible use in forecasting (Carniel and Di Cecca, 1999) or ture the global behavior of the system. A common method at least to highlight different regimes (Ripepe et al., 2002; to determine m is to use the first zero of the linear correla- Carniel et al., 2003; Harris et al., 2005; Jones et al., 2005). tion between the first and the last column of the trajectory By the use of spectral and dynamical analysis (Carniel et al., matrix. Another method used to determine m (Shaw, 2000) 2006) showed that the paroxysmal phase of 5 April 2003 uses the point where mutual information between the first have built up over at least the previous 2.5 hours, but the ap- and the last column of the trajectory matrix reaches the first plication of the Material Failure Forecast Method during the minimum (Fraser, 1986; Fraser and Swinney, 1986). This is days before the event revealed a consistent trend that sug- in principle a better choice, since it takes into account also gests a preparation of the paroxysm further in the past, an the non-linear correlation within the time series. evolution which was possibly temporarily “paused” the day before, before finally accelerating during the consistent dy- 2.1.2 Decomposition of time series namical phase of the last few hours. In any case, there is sufficient “persistence” in the data to look for some kind of Once the time series are embedded into the trajectory matrix precursor as a hind-cast exercise. In particular, the Singular X, the singular value decomposition is performed on such Spectrum Analysis is the methodology that shows the most matrix. The singular values decomposition technique allows promising potential in the search for precursors. We there- to obtain the decomposition: fore start by presenting this technique in further detail. X = USVT (1) where U is a (n−m+1) × (n−m+1) real orthogonal matrix, 2 The Singular Spectrum Analysis (SSA) technique V is a m×m real orthogonal matrix and S is a (n−m+1) ×m diagonal real matrix such that its elements are the singular Time series analysis has recently profited from the appli- values of the trajectory matrix X. This is done by computing cation of spectral decomposition of matrices (e.g. (Marelli the lagged covariance matrix et al., 2002; Mineva and Popivanov, 1996; Pereira and Ma- ciel, 2001)), which has its roots mostly in chaos theory (Tak- C = XT X (2) ens, 1981; Broomhead and King, 1981). The SSA con- sists of several steps (Golyandina et al., 2001; Aldrich and so C is a symmetric semidefinite positive matrix and then a Barkhuizen, 2003). In the first step (embedding), the one unique spectral decomposition exists: dimensional time series is recast as an L-dimensional time C = 838T (3) series (trajectory matrix). In the second step (singular value decomposition), the trajectory matrix is decomposed into a where 8 is a real orthonormal matrix such that its columns sum of orthogonal matrices of rank one. These two steps are the eigenvectors of C, and 3 is a real diagonal matrix 2 constitute the decomposition stage of SSA. In the third and such that its elements σi are the eigenvalues of C in decreas- fourth steps, the components are grouped and the time series ing order. One can now observe that associated with the groups are reconstructed. C = XT X 2.1 Decomposition stage T = USVT USVT 2.1.1 Embedding of time series = VST UT USVT Embedding expands the original time series into what is re- but U is an orthonormal matrix and S is a diagonal matrix, so ferred to as trajectory matrix of the system. This matrix is UT U=I and ST S=S2. Therefore associated with a certain window length that is the embed- 2 T ding dimension. The series is expanded by giving it a unitary C = VS V (4) lag and creating a certain number of lagged vectors. So, for Comparing Eqs. (3) and (4) one obtains that the time series x= {x1, x2, ... , xn} and for an embedding di- mension m, we obtain the trajectory matrix: 8 = V 3 = S2 x1 x2 ··· xm x2 x3 . xm+1 X = So one can deduce the important consequence that the right . .. . eigenvectors of X are the eigenvectors of C while the singu- xn−m+1 xn−m+2 . xn lar values of X are the square root of the eigenvalues of C. Nat. Hazards Earth Syst. Sci., 6, 903–909, 2006 www.nat-hazards-earth-syst-sci.net/6/903/2006/ R. Carniel et al.: Singular values decoupling 905 Therefore: and m σ1 0 ... 0 = X T 0 σ ... 0 noise X8i8i (10) 2 .